Hypoexponential Distribution with Different Parameters

(1)

Hypoexponential Distribution with

Different Parameters

Khaled Smaili1, Therrar Kadri2, Seifedine Kadry3

1_{Department of Applied Mathematics, Faculty of Sciences, Lebanese University, Zahle, Lebanon} 2_{Department of Mathematics, Faculty of Sciences, Beirut Arab University, Beirut, Lebanon}

3_{School of Engineering, American University of the Middle East, Eguaila, Kuwait} Email: [email protected], [email protected], [email protected]

Received January 27, 2013; revised March 4, 2013; accepted March 11, 2013

Copyright © 2013 Khaled Smaili et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

The Hypoexponential distribution is the distribution of the sum of n≥ 2 independent Exponential random variables. This distribution is used in moduling multiple exponential stages in series. This distribution can be used in many do-mains of application. In this paper we consider the case of n exponential Random Variable having distinct parameters. Using convolution, some properties of Laplace transform and the moment generating function, we analyse this case and give new properties and identities. Moreover, we shall study particular cases when



_i are arithmetic and geometric.

Keywords: Hypoexponential Distribution; pdf; Convolution; Laplace Transform; Moment Generating Function;

Expectation; Partial Fraction Expansion

1. Introduction

The Random Variable (RV) plays an important role in modeling many events [1,2]. In particular the sum of exponential random has important applications in the modeling in many domains such as communications and computer science [3,4], Markov process [5,6], insurance [7,8] and reliability and performance evaluation [4,5,9, 10]. Nadarajah [11], presented a review of some results on the sum of random variables.

Many processes in nature can be divided into sequen-tial phases. If the time the process spends in each phase is independent and exponentially distributed, then the overall time is hypoexponentially distributed. The service times for input-output operations in a computer system often possess this distribution. The probability density function (pdf) and cummulative distribution function (cdf) of the hypoexponential with distinct parameters were presented by many authors [5,12,13]. Moreover, in the domain of reliability and performance evaluation of sys-tems and software many authors used the geometric and arithmetic parameters such as [10,14,15].

In this paper we study the hypoexponential distribution in the case of n independent exponential R. V. with dis-tinct parameters

 

i  j

, n

for i j, written as



1 2

hypoexp  , ,



. We use in our work the prop-

erties of convolution, Laplace transform and moment

generating function in finding the derivative of the pdf of this sum and the moment of this distribution of order k. In addition, we deduce some new equalities re-lated to these parameters. Also we shall study the case when the parameters form an arithmetic and geometric sequence considered by [10,14,15] and find some new results.

th

k

2. Definitions and Notations

Let X X1, 2, , Xn be independent exponential random

variables with different respective parameters i ,

1, 2, ,

i  n, written as Xi Exp

 

i . We define the

random variable

1 n

n i i

S 



_ X

to be the Hypoexponential random variable with pa-rameters i, i1, 2, , n, written as



1 2



hypoexp ,

n n

S    , ,

Some notations used throughout the paper.

i

X : Exp

 

i .

n

S : hypoexp



 1, 2, ,Λn



.

X

f : The pdf of the random variable X.

X

F : The cdf of the random variable X.  k

: The thderivative of the k

X

(2)





L : Laplace-Stieltjes Transform. Therefore,



1

 _ _{: Laplace Inverse.} L

 

t

X : The moment generating function of X.

k

E X_ _: The moment of order k of the RV X.  : n₁

i

i product of all parameters.



i

P: _1, _1i_.

 

n j j i

j

  

 



i

 : n_1,





.

j i

j j i  



k

E :





1, ,



0 ; 1 ;1



, 0 0

n

n i i i

l Λl  l k



_l k  i n E  .

3. Applications on pdf and cdf Using Laplace

Th f the hypoexponential with distinct

pa-Transform

e pdf and cdf o

rameters were presented by many authors [2,7,11-13]. We shall state in thoerem 1 and propostion 1 these results and provide another proof using Laplace transform. Next, we give some new properties of its pdf, where new iden-tities are obtained.

Theorem 1. Let n2 and t0. Then

 

₁ i

 

n

n X

S i

i

f t t

P



and

f 



 

1 0, 

 

e

1 i

n

x n

S i

i

F x I

P  

 

 



x .

Proof. We have

 



i



i X

i

f x

s 

 



L ,

where smax

 

i for i1, 2, ,Λn. Since Xi are

independent then

 

n

S

f t is the co lutions o

i

X

nvo f f ,

1, 2, ,

i Λn written

 



1

n

S X

as



n

 

2

X X

f t  f f  Λ f t

and the Laplace transform of convolution of functions is the product of their Laplace transform, thus

 



_n



n₁



_i

 



S i X

f t f t

1 1

1

n _i n

i i

s s

 _

 

 

 



(1)

where



L L

 

max i .

s 

eorem [16], for

However, by Heaviside Expan-sion Th distinct poles gives that

 





1 ,

n _i

S_n i

i

A

f t 

s   _

where





L





1,

1

i n

j i

j j i

A

    





.

 

1 1

0, 1 e . n

i

n _i

S i

i

n t

i i

A f t

s

A  I t



 

 

  

 

 _ _



 





L

i i

i

A P



 

But . Thus

 

1

 

i

n X i

i

f

n

S

t P





.

On the other hand we have

f t 

 

1 0

n

S i

i

1 1

d

1 1 e _.

i

i i

x n X

x

n n

X

i i

f t t

F x

P P

 

 





 







But

P



then n₁1 1

i i

P

 



 

lim 1,

n

S

xF x  and we

con-clude that

 

e  

 

n

0, 1

1 i

n

x

S i

i

F x I x

P  

 



Next we shall discuss the derivative of 



. 

th

k

 

n

S

ning Pi form

f t

and many equalities are ob ed concer and some similar forms.

re tain

We start by noting from the p vious proof that

1

1 ₁

n i _P

i 



. Here, we shall state another simp proof le us

Proposi

ing Laplace transform.

tion 1. Let n2. Then

1

1 _1.

n i

i

P

 

Proof. We have from Equation (1),



 



Sn



i1 _s

n _i

i

f t 

  _



 

 _ _





L

where smax

 

i ,i1, 2, ,Λn. But from Theorem 1,

 

1

 

i n

n X

S i

f

i

t f t 



_

and

P

 







 







 

1

0,

1 .

i n

X n

S i

i

n _i

i

i i

f t f t

P

I t

P s

  

 







L L





1 1

n _i n _i

i i

s P s

 

i

  



 



 _  _

 





. Fors 0,

(3)

 

1 1

n _i

i



.

n _i

i

i Pi i



   

 





_  Therefore, 1

1 n

P

 1

i i





. 

Lemma 1. Let n2. Then

 

_{ }



n



1

n

k k i

S s i

i

f t

s

  



L  _ _



 

for .

Proof. The proof is done by induction. F 0  k n 1

or k0, we

from Equati (1)

have on

 



n



1

n _i

S

f t 

i

s 

 _

 

However, by Initial Value Theorem, we ave

 



_

_ _

L .

h

 



 



0 n n

t s

n 

 

 

1

lim lim

lim 0

S S

i i s

i

f t s f t

s

s 

 



 _ _



 



L

and for k1 we have  

_{ }







 



 





1

0

.

n n n

n

S S S

n _i

S i

k

f t s f t f

s f t s

s 

 



 

 

  _ _



 



L L

L

Moreover

n

k

in the same manner till the de-rivative, we obtain the result.

that  

_{ }



1





 

_{ }



 

₀

n n

k k

S S S

f  t s f t f 

L L

Continuing



1



th

n 

In the following propostion we shall prove the first



2



th

n derivative of the pdf of Sn are zeros,

which verifies the fact that the coefficient of variation of the hypoexponential distribution is less than one unlike the hyperexponential distribution that have the coeffi-cient of variation greater than 1.

Proposition 2. Let n2. Then

 

_{ }

Proof. Let , we have from Lemma 1,

0 Sn , if

t _k_n

0, if 0 2

lim k k n

f t      1

 

 2 n

 

_{ }



n



i1 i

s

n

k k i

S

f t 

 

 _ _



 



  L

fo and from Initial Value Theorem, we

have

r 0  k n 1

 

_{ }



 

_{ }



1

li lim lim

k k

n t

s

s f t 



0  L 

1

m

0, if 1 1

1 lim

, if 1 0

n n

k

S S

s s

n k s

f t

s n k

n k

s  

 

  

   

 _{ }

   

Corollary 1. Let Then



2

n .

 

1

0, if 1 1

1 , if

k n _i

n i

i

k n

P k n



  

   

 

 





Proof. We have

 

e i __0, _

 

i

t

X i

f t   I _ t

i

. Then the derivative of

th

r f_X is

 r

_{ }

r r1 it

i X

f t

 

1  e __0, _

 

i   I  t .

However, from Theorem 1,

 

₁ i

 

n

S

f n X

i i

f t t

P





,

then

   

 

_{ }

 

1 0,

 

₁ i

n

n X n

r

S i i

f t

P 1

e

1 i

r _r _t

r _i

i i

f t

I P   

 









 _

and

 

_{   }

1

1 0

lim _n 1 .

r

r n

r i

S i

t

i

f t

P

  

  



(2)

By Proposition 2, we obtain that

 

1 1

0, if 0 2

1 , if 1

r n _i

r i

i

r n

P r n



 



   

  _ _{ }





1

r

By replacing with we obtain

4. Applications on pdf and cdf Using

Moment Generating Function

In the previous section we saw the use of Laplace prop-erties in the proofs of the theorems and propositions. In a

si in th n

n-rating function to obtain more new related results. A new the

mo re , we

de-k the result. 

milar manner, is sectio we use the moment ge

form of the moment generating function of S_n and

ment of S_n of order k is given. Mo over

duce more new related equalities concerning P_i and

higher order derivatives of pdf of S_n.

 

1

 

i n

n X

S i

i

t t

P





 



.

Proof. We have

 

etSn

S

t E   f

  

_

tx

 

_x _x

  e

n

S _ _n d

and from Theorem 1,

 

1

 

i n

n X

S i

i

f t f t

P





,

then

 

1

 

n

S i i1

1 _e _d i

i

n tx n

X

i i

t x x

P

 

 . 

t f

P

 

(4)

Proposition 4. Let n2 and k0.Then 1

! n k

n i k

i i

k E S

P



  

 



Proof. We have from Proposition 3,

 

1

 

n

S i

  Xi .

i

t t

P





Then

 

1

d d

n i

S X

k i k

i

P

t 



 t

dk 1 dk

n

t t

 

and

 

1

0 0

d ₁ d

d d

n i

k k

n

S X

k i k

i

t t

t

P

t  t

 

 





t

which gives ₁

k i n k

n i

i

E X E S

P 

      

 



. But ik k!

i

k E X

   

  .

Thus we obtain the result. Next, we shall use the Proposition 3 and 4 to ind other identities on and higher orders for



f

i

P  

 

n

k S

f t

taking

. We start by noting that

 

0 1 and by

n

S

  t0 in

Proposition 3, we again obtain the result in Proposition 1,

that is n₁1 1

i i

P





 .

Proposition 5. Let n2 and k0. Then 1 2

1

1 2

1 1 _.

n k

n

k l l l

i

E

i i n

P   









_Λ where







1, , 0 ; 1 ;1

n

k n i i i

E  l Λl  l k



_l k in



.

Note that we may write

1 2

1 2 3

1 2

1 1

n

k k k

l l l

E   n I   i i i i





_Λ



_Λ , (3)

where







1, , 1



k k 1 2 k .

I  i Λi i    i Λ i n

However E_k and I_k are equivalent representing a

set of combination with repetition having n k 1 k  

 



  ies and , thus the above summation (3) shall be 1.

Proof. Let and . We have

and using multinomial expansion formula, we obtain possibilit E0I00

0

k n2



1 2



 

     

 k  k

E S_n E_ X X X_n _



1 2



1 2

1!2 !

k

E l l

!

! n

n

k

E X X X

l

n

l

l l

k n

E S   _{ }  _





Λ Λ .

Knowing that expectation is linear and Xi,

1, 2, ,

i  n are independent with

!

i i

l i

i l

l E X

i



  

  ,

then

1 2

! _.

k k

E S

n k

n l l l

E   _n

  _Λ

m Proposition 4,

  



(4)

Since fro

1

!

n k

n i k

i i

k E S

P



  

 



.

efore, Ther

1 2

1

1 2

1 1

n k

n

k l l l

i

E

i i n

P   

 



Λ . 

The following corollary is direct consequence of Proposition 5 and Equation (4), tak and 2 respectively.

Let Then

1)

ing k 0,1

Corollary 2. n2.

1

1 _1.

n i

i

P

 



2) n₁ 1 n₁1

i i

i i i

P 

  



and

 

n₁1.

n i

E S

i

 





3) ₁ ₂

1 i

i j

i i

P

1

n 1

j n i

 

 





and



2

1

2!

n

i j n i j

E S

       

 



.



In Pr 2,

e of

oposition we found the first deriva-

tiv





1th

n

n

S

f

Equati

at 0, However to find high vaties we recall on (2), that shows a direct ween

the th_derivative

er order deri relation bet

and ₁

k

n i

i i

P

 



k

n

S

f . Hence, in the

ne opostion we shall u ostion 5, to find an

eq

xt pr se Prop

uation for n₁ _i i

i

P

k

 



by finding a relation between





hypoexp  1, 2, ,Λn and

1 2 n

  

1 1 1

hypoexp , , , .

 Λ 

Proposition 6. Let n2 and kn. Then

 

1 1 2

1 2

1 .

k n

n i

E

1 n

l l

i

P

k

n

n     l

 



  



Λ

Proof. Let 2, 1,

i

 1, 2, ,

i

n   i Λn and



,



C_n hypoexp  ₁, ₂,Λ_n . The Theorem 1 f of Cn is

n by , the

pd

 

1

 

i n

n Y

C i

i

f t f t

B



(5)

where Yi Exp

 

_i and 1, 1

i n

i j j i j  B    



 .

, w find in

 

 

 

Next e shall P_i terms of B_i. We have

 





1, 1, 1 1, 1 1 1 1 ,

n i n j

i j j i j j i

j i n n j i i i P j j n                _  _          



m er i

  





ultiplying in the num ator and denom nator by

 

1, ,

n j j j i 



we obtain

 

1n1

i n i

i

P  B

 

  where

 

1 n

j j

 



_  . Hence, we may write

 

1

1 1 k n

i i

i i i

P    B

 

1n 1

k

n i n



 



.

But, for kn Proposition 5 gives that

1 2

1

1 2

k n

k n l l

i

E i Bi

 1 1 n n l n       Λ



. Therefore,

 

1 2 1 2 1 k n l l E i n n  1 2 1 1 1 2 1 1 1 . n n k n n k n _i l i l l l E P  n              





. Let and Then

 





Proposition 7 n2 k n 1.

 

_{   }

1 2

1

1 2 0

lim 1 n.

n

k n

k l l l

n S

t

E

f t    

   

  



Λ

Proof. We have from Equation (2),

 

_{   }

1

1 0

lim _n 1 .

k k n k i S i t i f t P      



and from Proposition 6,

 

1 2

1 1 2 1 k n i E i P 1 1

1 n.

k

n

n i l l l

n

  _ _{ } _



 



Λ

for Then,

  

1. k n

 

_{   }

1 2

1

1 2

lim 1 n.

n

k n

k l l l

n S

E

f t

0

t    

   

 



Λ

 

Many authors used the identity





1 1, 1 ₀ n n i j i j j i         _{ }  



 



and proved it in many long and complicated m s Here we shall submit a more simple prove. In addition,

w ties using the above

results.

Proposition 8. Let Then

ethod .

e shall find more related identi

2. n





1 1, 0. n i j i j j i       _{ }  _   





Proof. Let n2. By Corollary 1, taking i 1

1

n  

 we

have i n 1, then

1 0 n _i i i P   



. However,

 





1, n i j

n i n j j i

i P  _ _  





1 1 1, 1 1, 0 n i i j i

j j i

n n i

j i

j j i

              1 _ 









 Therefore,





1 1, 1 0. n n i j i j j i         _{ }     







Next we shall find a more general equality using our previous results.

. Let Then Proposition 9 n2.





 

1

0, if 0 2

1n k n      

   1 2

1 1,

1 2 n 1

k

n _i

n i

j i j j i

l l l k n                    





Proof. Let . Then, 1

, if

k n

n E 





 2 n

 









1, 1 1 1, 1 1 1, n k k i j

n i n j j i

n

i i

i j j i j i

k

n _i

n i

j i

j j i

P                       











(5) at

Suppose th 1 k n. We have from Corollary 1,

 

1

0, if 1 1

1 , if

k n _i n i i k n

P k n

            



and Equation (5) gives that





 

1

1 1

1,

0, if 1 1

n k n



  _   

 _{  }



ik ₁ _{, if}

n n

i

j i

j j i k n

          _   





(6)

the case when k n 1,where 1 2 0

1 2 n 1

l

l l

n E

   



Λ .

position 6, Now, suppose kn. By Pro

 

1 2

1 2 n

k n

l

l l

n 1

k

n n i

1 1

i

E i

P     

Λ

   



and the Equation (5) gives that





 

1 2

1

1 2 1

1,

1 n

k

n

n i l l l

n i

i j



k n

E n

j j i

  

 

 



 _{  }

 

 





_  Λ .

 



case

w

5. The Main Results

llowing

Theorem 2. Let Then

1)



e last

nd 7 in the fo

1 n ,

0. Then Also, replace by k we obtain th hen n k 1.

1 k



We summarize Proposition 2 a theorem.

2.

n

 

_{ }

 

0

1 2

lim _nk S t

k n

f t







   1 1 2

0, if 0 2

if 1

n

l l l

k n k n

   



   

 



Λ

1

k n

E 



Also Corollary 1 and Proposition 5 and 6 can be summarized in the following theorem.

Theorem 3. Let n2 and k

 

1 2

if 0

1 l l ln,

n k n

 

 

 1  

1

1 if

k n

k

n _i

n i

i

E

k n

P 

 

  

 

  



_

and

2)

0, 

1 2

1

1 2

k

E

i i n

1 1

n

l l

i

Pk   ln.

 



Λ

We recall Propostion 9 in the following corollary of Theorem 3.

Corollary 3. Let n2. Then





 

1 2

1

k n

E 



6. Case of

1 1,

1

1 2

0,

1

k

n _i

n i

j i j j i

n l l

  

  

 



 

 

  

 

 

  _









Λ

metic

The study of reliability and p tion of

systems and softwares use in dent

exponential R.V. with distinct del of

Jelinski and Moranda [14], con

rame-ters changes in an arithmetic

if 0 2

1 ln, if

n

k n k n



  

 

eometric

erformance evalua general sum of indepen

parameters. The m sidered that the pa sequence i i

Arith

and G

Parameters

o

1 d

  nsidered the model

  .

when

i

 changes in an geometric sequence  i i1r

tion, we study t

. In this he hypoexponential in cases

n

sec these two

,

when the parameters are arithmetic and geometric, and we present their pdf.

6.1. Case of Arithmetic Parameters

We first consider the case when i,i1, 2,Λ n

arithmetic sequence of common difference d.

Lemma 2. For all 1 i n.

form a

 

  

₁ 1 1 !



1_.

1

i n

i

n d n

 _{ }   



 

1 i  _ 

 

Proof. Suppose that

Moreover, Moranda [15], co

i

 form an arithmetic se



j    



i

quence of common difference d. Then _j i i d



. We

have





1, n

i j j i j

 

_

_ _   .

Hence,





 



 





1 2 1

i i i i i n

d

 







   





  

1

 



₁

1 2 2

1i 1 ! ! _n

i d d d n i d

i n i

1

i i i

        _  _    

 _

    

   

 

However,



 





1 !₁



1 

     . !

1 ! ! n n

i n i

n n

i

i i

   

         Then

  

₁ 1 1 !



1_,

1 1

i n

i

n d n i

 _{ }   



 

 _ 

 



Lemma 3. For all 1 i n.

 

1 1  1

n

i n i

  

 

  .

e have from Lemma 2, Proof. W

  

₁ 1 1 !



1

1 1

i n

i

n d n

i

 _{ }   



 

 _ 

 

for all 1 i n. Replace i by n 



i 1



, we obtain

 





 

  



 

1 1

1 ! 1

1

n i

n 1 1 !

1 1 .

1 1

n i

n i n n

i

n

d

n i

n d n i

 n



 

  

  

 

 



   



 

 _ 

 

(7)

Thus we obtain the resu t. l 

 

1 0, 

 

e i n

t n

S i

i

f t I

 

 

 





t ,

where

  



 

1 ₁ 1

1

1 !

1 1

1 1 i

 

i n n

i n i

n d n

  _  

  

   



 

 

fo

Proof. We have from Theorem 1 r all 1 i n.

 





 

0, 1

1,

0, 1

1,

e

1

e

,

i n

i

t

n _i

S i

n _i

j j i j n t

i j

n j j i

n i

j i

j j i

f t I t

I t



 

  

 

 



 

 

  

 



 

 













that can be written as

 

1 0, 

 

e i n

t n

S i

i

f t I

 

 

 





t ,

where 1,



i



n

i j j i j

 



_ _   obtain the result.

and by the Lemm 2 and 3 we

6.2. Case of Arithmetic Parameters

ase when

as



Next, we consider the c  i,i 1, 2, ,Λn r.

fo geometric sequence of common ratio

Proposition 11. Let Then

rm a

2. n

 





 

1

e i n

t

n i

S i n 0,

1, 1

i j j j i

f t I

   







Proof. We have from Theorem 1,

t

r 

  

.

 

1 0, 

 

1, 1

n

S i

n _i

j j i

e it

n _i

j

f t

 

  

  





I t

   

.

Suppose now the parameter   

 

i

 en

form geometric se-quence of common ratio . Thr i jri j d



 an





1, 1

n _i n

i j j i j

j P



  

 _  _

 



1, 1

i j j i r

 



 

 . 

We may also note that the equalities obtaine for represent here a special case and worth mentioning s as

d Pi

uch





1

1 _1.

1

n n

i i

j j i r

 

 

 





1,

j

7. Conclusion

The pdf and cdf and some related properties of he hypoexponential distribution with distinct parameters

w ng

Also with the help of some known computational theorems as Heaviside expansion theorem and multino-mial expansion formula the kth order der ative of

t

ere established. The proofs have been done by usi Laplace transform and moment generating function tech-nique.

iv fS_n

es-lities.

i

and the moment of this distribution of order k were tablished, in addition for some new related equa

f for models when the parameters

Eventually, the pd 

are arithmetic and geometric were presented. However the other two cases for hypoexponential distribution when the parameters are equal or not all equal can be studied and observed for future studies. It may be checked if they have the same properties as in this paper.

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